Discovering Non-Standard Semantics of Semi-Stable Attributes
Angelina A. Tzacheva and Zbigniew W. Ras
University of North Carolina – Charlotte
College of Information Technology, Computer Science Department
Charlotte, NC 28223, USA
aatzache@uncc.edu and ras@uncc.edu
Abstract
A new class of rules, called action rules, show what actions
should be taken to improve the profitability of customers. Action
rules introduced by (Ras and Wieczorkowska, 2000a) and
investigated further by (Ras and Gupta, 2002a) assume that
attributes in a database are divided into two groups: stable and
flexible. These reflect the ability of a business user to influence
and control their change for a given consumer. In this paper, we
introduce a new classification of attributes partitioning them into
stable, semi-stable, and flexible. Values of stable attributes can
not be changed for a given consumer (for instance “maiden
name” is an example of such an attribute). So, stable attributes
have only one interpretation. If values of an attribute change in a
deterministic way as a function of time (for instance values of
attribute “age” or “height”), we call them semi-stable. All
remaining attributes are called flexible. Clearly, in the process of
action rule extraction, stable attributes are highly undesirable.
What about semi-stable attributes? Although, they seem to be
quite similar to stable attributes, the difference between them is
quite essential. Semi-stable attribute may have many different
interpretations but among them only one interpretation is natural
and it is called standard. All its other interpretations are called
non-standard. In a non-standard interpretation, a semi-stable
attribute can be classified as flexible. In a single database we
may easily fail to identify attributes which have non-standard
interpretation. In this paper, we show how distributed
information system introduced by (Ras, 2000b, 2001a) can be
used as a tool to identify which semi-stable attributes have nonstandard interpretation so they can be classified as flexible. This
way, by decreasing the number of stable attributes in a database
we may discover action rules which would not be discovered
otherwise.
Introduction
In (Ras and Wieczorkowska, 2000a) the notion of special
type of rules, called action rules, was introduced. These
rules can be constructed from classification rules to suggest
a way to re-classify objects (for instance customers) to a
desired state. In e-commerce applications, this reclassification may mean that a consumer not interested in a
certain product, now may buy it, and therefore may fall
into a group of more profitable customers.
Copyright © 2003, American Association for Artificial Intelligence
(www.aaai.org). All rights reserved.
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FLAIRS 2003
These groups are described by values of classification
attributes in a decision table schema. In paper by (Ras and
Wieczorkowska, 2000a), all attributes are divided into
stable and flexible. This time, a new subclass of attributes
called semi-stable attributes is introduced. Semi-stable
attributes are typically a function of time, and undergo
deterministic changes (for instance attribute “age” or
“height”). Different interpretations, called non-standard, of
such attributes may exist, and in such cases all these
attributes can be treated the same way as flexible attributes.
In the algorithm of action rule extraction, proposed by (Ras
and Wieczorkowska, 2000a) attributes which are not
flexible are highly undesirable. By identifying which semistable attributes have non-standard interpretation, we
increase the number of flexible attributes and the same may
increase the confidence of generated action rules.
Assuming that attribute is flexible, we may find a way to
change its value for a given object. However, quite often,
such a change cannot be done directly to a chosen attribute
(for instance to the attribute “profit”). In that situation,
definitions of such an attribute in terms of other attributes
have to be learned. These definitions are used to construct
action rules showing what changes in values of attributes,
for a given consumer, are needed in order to re-classify this
consumer the way business user wants. In a distributed
system, we may search for definitions of these flexible
attributes looking at either local or remote sites for help.
The application of semi-stable attributes to the process of
action rules mining involves detection of nonstandard
interpretations of semi-stable attributes. At local system
level detection is possible, but limited to dependencies
existing between local attributes. At distributed
information systems level the detection of nonstandard
interpretations involves discovering semantic
inconsistencies, addressed by (Ras and Dardzinska,
2002b).
Information Systems and Decision Tables
An information system is used for representing business
knowledge. (Pawlak, 1985a) gives the following definition:
By an information system we mean a pair S = (U, A),
where:
1. U is a nonempty, finite set of objects (called customer
identifiers),
2. A is a nonempty, finite set of attributes i.e. a:U Æ Va
for a ΠA, where V a is called the domain of a.
Information systems can be seen as generalizations of
decision tables (Pawlak, 1985a). Partition of the set of
attributes into conditions and decisions is given in any
decision table. We assume that the set of conditions is
partitioned into stable, semi-stable, and flexible conditions.
Attribute aŒA is called stable for the set U if its values
assigned to objects from U can not be changed by a
business user. An attribute is called semi-stable, if it is a
function of time and it is changing, in its standard
interpretation, in a deterministic way. Otherwise, it is
called flexible. Date of birth is an example of a stable
attribute. Age is an example of semi-stable attribute (its
value “young” in a non-standard interpretation may mean
“behaving as a young person”). Interest rate on any
customer account is an example of a flexible attribute. For
simplicity reason, we will consider decision tables with
only one decision. We adopt the following definition of a
decision table:
By a decision table we mean any information system of
the form S = (U, A1 » A2 » A3 » {d}), where d
œ A1 » A2 » A3 is a distinguished attribute called the
decision. Elements of A1 are called stable conditions, the
elements of A 3 are called semi-stable, and A 2 » {d} are
called flexible conditions.
The assumption that attribute d is flexible is quite
essential. Otherwise we would be unable to re-classify
objects in U from the point of view of attribute d. So, if d
is flexible and we want to change its value for a given
object, values of some attributes from A 2 » A3 have to
be changed as well.
Before we proceed, certain relationships between values
of attributes from A 2 and A3 and values of the attribute d
have to be presented first.
Action Rules
(Ras and Wieczorkowska, 2000a) proposed a method to
construct action rules from a decision table containing
both stable and flexible attributes. In this section, let us
assume that for each attribute in A3 we know its semantics
and also we know if it is stable or flexible. So A3 = ∅,
which means some semi-stable attributes are moved to A1,
some to A2.
Assume now that for any two collections of sets X, Y,
we write, X Õ Y if ("xŒX)($yŒY)[ xÕ y ]. Let S = (U,
A1 » A2 » {d}) be a decision table and B Õ A 1 » A2.
We say that attribute d depends on B if CLASSS(B) Õ
CLASSS(d), where CLASSS(B) is a partition of U
generated by B (Pawlak, 1985a).
Assume now that attribute d depends on B where B Õ
A1 » A2. The set B is called d-reduct in S if there is no
proper subset C of B such that d depends on C. The
concept of d-reduct in S was introduced, in rough sets
theory (Pawlak, 1985a), to identify minimal subsets of
A1 » A2 such that rules describing the attribute d in terms
of these subsets are the same as rules describing d in terms
of A1 » A2. It was shown that in order to induce rules in
which THEN part consists of the decision attribute d and
IF part consists of attributes belonging to A1 » A2, only
subtables (U, B » {d}) of S where B is a d-reduct in S
can be used for rules extraction.
By L(r) we mean all attributes listed in IF part of rule r.
For example, if r = [ (a1,3)*(a2,4) Æ (d,3)] is a rule then
L(r) = {a1,a2}. By d(r) we denote the decision value of a
rule r. In our example d(r) = 3. Similarly, a1(r)=3.
If r1, r2 are rules and B Õ A1 » A2 is a set of
attributes, then r1/B = r2/B means that the conditional
parts of rules r1, r2 restricted to attributes B are the same.
For example if r1 = [(a1,3) Æ (d,3)], then r1/{a1} =
r/{a1}.
Algorithm for constructing action rules, implemented as
system DAR was given by (Ras and Wieczorkowska,
2000a).
For each pair of rules (r1, r2) satisfying the conditions
r1/A1 = r2/A1, d(r1)=k1, d(r2)=k2 where k1< k2,
if (b1, b2,…, bp) was a list of all attributes in L(r1) «
L(r2) « A 2 on which r1, r2 differ and r1(b1)= v1, r1(b2)=
v2,…, r1(bp)= vp, r2(b1)= w1, r2(b2)= w2,…, r2(bp)= wp
then the algorithm DAR generates the following (r1,r2)action rule:
If [(b1, v1Æ w1) Ÿ (b2, v2 Æ w2) Ÿ…
Ÿ (bp, vp Æ wp)](x) then the ranking profit of
customer x is expected to change from k1 to k2.
Object x supports this action rule if it satisfies the
properties: b1(x)=v1, b2(x)=v2, ….bp(x)=vp, d(x)= k1.
By the support of an action rule r we mean all the objects
supporting that rule. It is denoted by Sup(r).
If the change of values of attributes of object x ΠSup(r)
will match the term
[(b1, v1Æ w1) Ÿ (b2, v2 Æ w2) Ÿ…Ÿ
(bp, vp Æ wp)](x)
and the resulting object is either classified in S as k2 or
not classified at all, then the action rule r successfully
suports x. The set of objects successfully supported by r is
denoted by SSup(r). By the confidence of rule r we mean
Conf(r) = SSup(r)/ Sup(r).
Example 1. Assume that S = ({x1,x2,x3,x4,x5,x6,x7,x8},
{a,c}»{b}» {d}) is a decision table represented as Table
1. The set {a,c} contains stable attributes, b is a flexible
attribute and d is a decision attribute.
It can be easily checked that {b,c}, {a,b} are the only
two d-reducts in S.
Applying, for instance, LERS discovery system
(Chmielewski and Grzymala-Busse, 1993a) the following
definitions are extracted from S:
(a,0) Æ (d,L),
(c,0) Æ (d,L),
(b,R) Æ (d,L),
(c,1) Æ (d,L),
(b,P) Æ (d,L), (a,2)Ÿ(b,S) Æ (d,H),
(b,S)Ÿ(c,2) Æ (d,H).
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331
Now, let us assume that (a, v Æ w) denotes the fact that
the value of attribute a has been changed from v to w.
Similarly, the term (a, v Æ w)(x) means that a(x)=v has
been changed to a(x)=w. Saying another words, the
property (a,v) of object x has been changed to property
(a,w).
x1
x2
x3
x4
x5
x6
x7
x8
a
0
0
0
0
2
2
2
2
b
S
R
S
R
P
P
S
S
c
0
1
0
1
2
2
2
2
d
L
L
L
L
L
L
H
H
Table 1
If we take r = [(b, R Æ S) fi (d, L Æ H)] as the action
rule, then Sup(r)={x2,x4}, Conf(r) = 1.
Semi-Stable Attributes
The notion of action rules introduced by (Ras and
Wieczorkowska, 2000a) divides attributes into two groups:
stable and flexible. These reflect the ability of a business
user to influence and control their change for a given
consumer. In the process of action rule extraction stable
attributes are highly undesirable. We introduce a new
classification of attributes into stable, semi-stable, and
flexible taking into consideration semantics of atributes
which clearly may differ from database to database.
Value of a stable attribute a for a given object cannot be
changed by a business user in any interpretation of a. All
such interpretations are called standard.
An example of such an attribute is “date of birth”.
Standard interpretations of this attribute may differ in a
granularity level. It is possible that one stable attribute
implies another one.
There is a special subset of attributes called semi-stable,
which at first impression may look stable, but they are a
function of time and undergo changes in a deterministic
way. Therefore, they cannot be called stable. The change is
not necessarily in a linear fashion (see Graph 1). An
attribute may be stable for a period of time, and then begin
changing in certain direction as shown on Graph 2.
Semi-stable attributes may have many interpretations,
some of which might be nonstandard. We denote by M s(a)
the set of standard interpretations of attribute a and by
Mn(a) the set of non-standard interpretations of a. If the
attribute a has a nonstandard interpretation, I(a) ΠMn(a),
then it can be changed, and thus it may be seen as a flexible
attribute in action rule extraction.
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Va
x(t)
Graph 1. Semi-stable attribute “Age”
For instance, if a = “age” and Dom(a) = {young,
middle-aged, old}, the author of the database may indeed
input young for a person who behaves as young when their
actual age is middle-aged. Then the interpretation is
nonstandard. The business user can therefore influence this
attribute. For example, if the following action rule was
mined for object x
r1 = [[(a, young Æ middle-aged) ](x) fi
[(d, L Æ H)](x)]
with respect to decision attribute d (ex. loyalty) the
business user would like to change the attribute value
“young” to “middle-aged” for object x. Since the database
author interpretation is nonstandard related to the behavior
associated with certain age, if the object is put into special
conditions that can affect its behavior, such as top
university, the attribute value can be changed, and the same
object x might be re-classified from low loyalty to high
loyalty.
Va
x(t)
Graph 2. Semi-stable attribute “Height”
Many cases of nonstandard interpretations could be found
in databases. It is particularly important for those to be
detected when mining for global rules in distributed
knowledge systems. An example is the attribute “height”.
Consider the following situation: Chinese people living in
the mountains are generally taller than majority of Chinese
population. If Dom(a) = {short, medium, tall} for attribute
“height”, and a system S1, contains data about Chinese
population in the mountains. The author of the database
may consider a certain Chinese person living in the
mountains medium height in relation to the rest. Now
assume another system S2 containing data about Chinese
people living in popular urban area. In global action rule
extraction, if S2 is mined for rules, the interpretation would
regard the height value medium from S1 as tall. Therefore,
the interpretation in S1 is nonstandard.
Numeric attributes may possess nonstandard
interpretations as well. Consider for instance the attribute
“number of children”. When one is asked about the number
of children one has, that person may count step-children, or
children who have died. In such a case, the interpretation is
nonstandard.
A flexible attribute is an attribute which value for a given
object varies with time, and can be changed by a business
user. Also, flexible attributes may have many
interpretations. “Interest rate” is an example of a flexible
attribute.
Assume that S = (U,A) is an information system which
represents one of the sites of a distributed information
system (DIS). Also, let us assume that each attribute in A is
either stable or flexible but we may not have sufficient
temporal information about semi-stable attributes in S to
decide which one is stable and which one is flexible. In
such cases we will search for additional information,
usually at remote sites for S, to classify uniquely these
semi-stable attributes either as stable or flexible.
Discovering Semantic Inconsistencies
Different interpretations of flexible and semi-stable
attributes may exist. Semi-stable attributes, in a nonstandard interpretation, can be classified as flexible
attributes and therefore can be used in action rule
extraction. We discuss a detection method of nonstandard
interpretations of a semi-stable attribute at local
information system level, and next at distributed
information systems level.
Detection of a nonstandard interpretation at local level is
limited to the dependency of one semi-stable attribute to
another semi-stable attribute for which it is known that its
interpretation is standard. Attribute related to time must be
available in the information system, such as the attribute
“age”. Furthermore, information about certain breakpoints
in attribute behavior is required, such as the break points
shown in Graph 2. This information can be placed in the
information system ontology.
Assume that S=(U,A) is an information system and I is
the interpretation used for attributes in S. Also, assume that
both a ,b ΠA are semi-stable attributes, I(a) ΠMs and the
relation P {a, b}(S) Õ {(va , vb): v a Œ Dom(a), vb Œ
Dom(b)} is obtained by taking projection of the system S
on {a,b}. The ontology information about break points for
attributes a and b in S, represented in the next section as
relation RI(a),I(b), is assuming that the interpretation I is
standard for attribute b. It is possible that some tuples in
P {a, b}(S) do not satisfy the break points requirement given.
In such a case the interpretation of B is nonstandard, I(B) Œ
Mn.
Consider the following situation:
Example 3. Assume that S1 = (U1, {a}»{h, j}»{b}) is
an information system represented by Table 2, where {a} is
a set of stable attributes, {h, j} is a set of semi-stable
attributes, and {b} is a set of flexible attributes, where h is
“height” and j is “number of cousins”. The interpretation
of j is known to be standard, I(j) ΠMs. The system
represents a local site.
a
0
0
0
0
2
2
2
2
x1
x2
x3
x4
x5
x6
x7
x8
h
a
b
b
c
b
b
a
c
b
S
R
S
R
P
P
S
S
j
m
m
n
m
n
n
m
m
Table 2
Graph 3 shows the break points defined by the system’s
ontology for attributes h and j as a function of time t. The
number of cousins grows as the height grows, since the
person is young, and the parents’ brothers and sisters have
newborn children. The number of cousins decreases, as the
height becomes constant or shrinks, since for a person who
is middle-aged or old, the number of his/her cousins
naturally decreases as they die. Therefore,
if I(h) ΠMs and I(j) ΠMs,
then RI(h),I(j) = {(a, m), (b, m), (b, n), (c, n)}
is placed in the ontology layer for system S1.
b
c
h
a
m
n
j
x(t)
Graph 3. Dependency relation between
attributes h and j assuming standard
interpretations for both of them.
From Figure 3, we see that relation instance
(c,m)ŒP {h,b}(S1) representing objects x4, x8 does not
belong to R I(h),I(j). Therefore, I(h) ΠMn
In other words, objects x4, x8 do not satisfy the break
point requirement given on Graph 3, thus the interpretation
of attribute “height” is nonstandard.
Single information systems provide limited capability of
detecting nonstandard semantics. Distributed information
systems supply greater ability to detect nonstandard
semantics. They also give the opportunity to business users
to seek alternative solution at remote sites. This is
particularly important in a situation when they are either
not willing or not able to undertake the suggested actions
from their local site. In a distributed information systems
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333
(DIS) scenario semantic inconsistencies can be detected
even if temporal information is not available. With large
number of sites containing similar attributes in DIS, certain
trends can be observed, such as association rules with high
confidence and support common for all the sights. In such
a situation, it is also possible that a small number of sites
do not support those common rules, or even contradict
them. This case presents a hint for nonstandard attribute
interpretation, and semantic inconsistencies.
Assume that S, S1, S 2, … , Sn where S i = (Ui , B i ),
(i=1,2,…,n) are information systems which are parts of
DIS. Association rule mining is performed on all systems
in DIS. Rules satisfying the minimum support ms and
minimum confidence mc thresholds are mined. If a rule
[w a * w b * … * w c ] Æ wd [ms, mc] (1)
where d ΠA is a semi-stable attribute in S = (U,A) is
extracted from S and supported by many sites in DIS, it is
called a trade, or a common rule for DIS. If rule (1) is
supported either only by S or by a very small number of
sites in DIS and at the same time rule (2) is supported by
many sites in DIS
ÿ([wa * w b * … * w c ] Æ wd ) [ms, mc] (2)
then the attribute d has nonstandard interpretation in S.
Assume that we do not know if the interpretation of a
semi-stable attribute dŒB1 at site S 1 = (U1, B 1) is standard.
We have discussed the case when attribute d is defined in
S 1 in terms of a semi-stable attribute b, which has standard
interpretation in S1. Namely, if we identify an object in S 1
which description contradicts information about attributes
d, b stored in S1 ontology, then attribute d has non-standard
interpretation in S1. However, it can happen that we do not
have any information about the interpretation of b in S1.
We can either look for a definition of d in terms of another
semi-stable attribute in S1 or look for a definition of d in
terms of attribute b at another site of DIS. If we cannot find
any attribute other than d, which is semi-stable and has
non-standard interpretation in S1, we contact another site.
Let B1ss be the set of all semi-stable attributes in S 1. We
search for sites Si such that d Œ Bi and Bi ss « B 1ss ≠ ∅. Let
Id be the collection of such sites and b Œ B1ss « Bi ss , where i
ΠId.
In the case where, the interpretation of both attributes d,
b is standard, if I Œ Id satisfies the property that any b Œ
B 1ss « B i ss has standard interpretation in Si , then i is not
considered. Thus, we need to observe another site from Id.
This algorithm was tested on a DIS consisting of thirteen
sites representing thirteen Insurance Company Datasets,
with a total of 5000 tuples in all DIS sites. Semi-stable
attributes with non-standard interpretation have been
detected and used jointly with flexible attributes in action
rules mining. The confidence of these action rules is
usually higher than the confidence of the corresponding
action rules based only on flexible attributes.
Now let us assume that I {d, b} is the set of sites in Id such
that b Œ Bi « B 1ss . We extract rules at sites I{d, b} describing
d and having b on their left side. Either association rules
discovered at site S1 will support association rules
discovered at majority of sites I{d, b} or conflict many of
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them. We claim that the interpretation of attribute d is
standard in the first case. In the second case, it is nonstandard.
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